Difference between revisions of "2020 IMO Problems/Problem 5"

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Problem 5. A deck of n > 1 cards is given. A positive integer is written on each card. The deck
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==Problem ==
has the property that the arithmetic mean of the numbers on each pair of cards is also the geometric
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A deck of <math>n > 1</math> cards is given. A positive integer is written on each card. The deck has the property that the arithmetic mean of the numbers on each pair of cards is also the geometric mean of the numbers on some collection of one or more cards.  
mean of the numbers on some collection of one or more cards.
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For which n does it follow that the numbers on the cards are all equal?
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For which <math>n</math> does it follow that the numbers on the cards are all equal?
  
 
== Video solution ==
 
== Video solution ==
 
https://www.youtube.com/watch?v=dTqwOoSfaAA [video covers all day 2 problems]
 
https://www.youtube.com/watch?v=dTqwOoSfaAA [video covers all day 2 problems]

Revision as of 11:25, 14 May 2021

Problem

A deck of $n > 1$ cards is given. A positive integer is written on each card. The deck has the property that the arithmetic mean of the numbers on each pair of cards is also the geometric mean of the numbers on some collection of one or more cards.

For which $n$ does it follow that the numbers on the cards are all equal?

Video solution

https://www.youtube.com/watch?v=dTqwOoSfaAA [video covers all day 2 problems]

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